Archive for Mathematical Logic

, Volume 56, Issue 3–4, pp 187–213 | Cite as

Chains of saturated models in AECs

Article

Abstract

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove:

Theorem 0.1.IfKis a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough\(\lambda {:}\)
  1. (1)

    The union of an increasing chain of\(\lambda \)-saturated models is\(\lambda \)-saturated.

     
  2. (2)

    There exists a type-full good\(\lambda \)-frame with underlying class the saturated models of size\(\lambda \).

     
  3. (3)

    There exists a unique limit model of size\(\lambda \).

     
Our proofs use independence calculus and a generalization of averages to this non first-order context.

Keywords

Abstract elementary classes Forking Independence calculus Classification theory Stability Superstability Tameness Saturated models Limit models Averages Stability theory inside a model 

Mathematics Subject Classification

Primary 03C48 Secondary 03C47 03C52 03C55 03E55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albert, M.H., Grossberg, R.: Rich models. J. Symb. Log. 55(3), 1292–1298 (1990)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Baldwin, J.T.: Categoricity, University Lecture Series, vol. 50, American Mathematical Society (2009)Google Scholar
  3. 3.
    Boney, W., Grossberg, R.: Forking in short and tame AECs. Ann. Pure Appl. Log. (2017). doi:10.1016/j.apal.2017.02.002
  4. 4.
    Boney, W., Grossberg, R., Kolesnikov, A., Vasey, S.: Canonical forking in AECs. Ann. Pure Appl. Log. 167(7), 590–613 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Boney, W.: Tameness and extending frames. J. Math. Log. 14(2), 1450007 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Boney, W.: Tameness from large cardinal axioms. J. Symb. Log. 79(4), 1092–1119 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Drueck, F.: Limit models, superlimit models, and two cardinal problems in abstract elementary classes, Ph.D. thesis (2013). http://homepages.math.uic.edu/~drueck/thesis
  8. 8.
    Grossberg, R., Lessmann, O.: Shelah’s stability spectrum and homogeneity spectrum in finite diagrams. Arch. Math. Log. 41(1), 1–31 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Grossberg, R.: A Course in Model Theory I (in preparation)Google Scholar
  10. 10.
    Grossberg, R.: On chains of relatively saturated submodels of a model without the order property. J. Symb. Log. 56, 124–128 (1991)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Grossberg, R.: Classification theory for abstract elementary classes. Contemp. Math. 302, 165–204 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Grossberg, R., VanDieren, M.: Galois-stability for tame abstract elementary classes. J. Math. Log. 6(1), 25–49 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Grossberg, R., VanDieren, M., Villaveces, A.: Uniqueness of limit models in classes with amalgamation. Math. Log. Q. 62, 367–382 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Harnik, V.: On the existence of saturated models of stable theories. Proc. Am. Math. Soc. 52, 361–367 (1975)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Makkai, M., Shelah, S.: Categoricity of theories in \({L}_{\kappa,\omega }\), with \(\kappa \) a compact cardinal. Ann. Pure Appl. Log. 47, 41–97 (1990)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Shelah, S.: The lazy model theoretician’s guide to stability. Log. Anal. 18, 241–308 (1975)MathSciNetMATHGoogle Scholar
  17. 17.
    Shelah, S.: Classification of non elementary classes II. Abstract elementary classes. In: Baldwin, J.T. (ed.) Classification Theory (Chicago, IL, 1985), Lecture Notes in Mathematics, vol. 1292, pp. 419–497. Springer, Berlin (1987)Google Scholar
  18. 18.
    Shelah, S.: Classification theory and the number of non-isomorphic models. In: Studies in Logic and the Foundations of Mathematics, vol. 92, 2nd edn. North-Holland, Amsterdam (1990)Google Scholar
  19. 19.
    Shelah, S.: Categoricity for abstract classes with amalgamation. Ann. Pure Appl. Log. 98(1), 261–294 (1999)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Shelah, S.: Classification Theory for Abstract Elementary Elasses, Studies in Logic: Mathematical Logic and Foundations, vol. 18. College Publications, Norcross (2009)Google Scholar
  21. 21.
    Shelah, S.: Classification Theory for Abstract Elementary Classes 2, Studies in Logic: Mathematical Logic and Foundations, vol. 20. College Publications, Norcross (2009)Google Scholar
  22. 22.
    Shelah, S., Villaveces, A.: Toward categoricity for classes with no maximal models. Ann. Pure Appl. Log. 97, 1–25 (1999)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    VanDieren, M.: Categoricity in abstract elementary classes with no maximal models. Ann. Pure Appl. Log. 141, 108–147 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    VanDieren, M.: Erratum to “Categoricity in abstract elementary classes with no maximal models” [Ann. Pure Appl. Logic 141 (2006) 108–147]. Ann. Pure Appl. Log. 164(2), 131–133 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Vasey, S.: Building independence relations in abstract elementary classes. Ann. Pure Appl. Log. 167(11), 1029–1092 (2016)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Vasey, S.: Forking and superstability in tame AECs. J. Symb. Log. 81(1), 357–383 (2016)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Vasey, S.: Infinitary stability theory. Arch. Math. Log. 55, 567–592 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations